If you're responsible for your own actions, you are accountable for them. You are accountable for your behavior in school. So even if your friend throws pudding at you at lunch, you don't have the right to fling mystery meat back.
Accountable comes from the word "account," which is itself from the Old French word acont, meaning to count (as in money). While you can be accountable for monetary damages, you're most likely to be held accountable for your actions and behavior, and the actions of your children and pets. If your dog bites another dog, you may be accountable for any vet bills incurred by the other dog's owner.